Problem: Simplify; express your answer in exponential form. Assume $t\neq 0, q\neq 0$. $\dfrac{{t^{4}q^{5}}}{{(t^{-5}q^{3})^{-1}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${t^{4}q^{5} = t^{4}q^{5}}$ On the left, we have ${t^{4}}$ to the exponent ${1}$ . Now ${4 \times 1 = 4}$ , so ${t^{4} = t^{4}}$ Apply the ideas above to simplify the equation. $\dfrac{{t^{4}q^{5}}}{{(t^{-5}q^{3})^{-1}}} = \dfrac{{t^{4}q^{5}}}{{t^{5}q^{-3}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{4}q^{5}}}{{t^{5}q^{-3}}} = \dfrac{{t^{4}}}{{t^{5}}} \cdot \dfrac{{q^{5}}}{{q^{-3}}} = t^{{4} - {5}} \cdot q^{{5} - {(-3)}} = t^{-1}q^{8}$